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If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig ...
An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A short exact sequence is a bounded exact sequence in which only the groups A k, A k+1, A k+2 may be nonzero. For example, the following chain complex is a short exact sequence.
Alternatively, one can argue that every short exact sequence of k-vector spaces splits, and any additive functor turns split sequences into split sequences.) If X is a topological space , we can consider the abelian category of all sheaves of abelian groups on X .
The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter ...
S (k,k) is the polynomial ring R; this is an example of Koszul duality. By the general properties of derived functors, there are two basic exact sequences for Ext. [6] First, a short exact sequence of R-modules induces a long exact sequence of the form
The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology .
If any of these statements holds, the sequence is called a split exact sequence, and the sequence is said to split. In the above short exact sequence, where the sequence splits, it allows one to refine the first isomorphism theorem, which states that: C ≅ B/ker r ≅ B/q(A) (i.e., C isomorphic to the coimage of r or cokernel of q) to:
The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences. ... One easy example of local homology is ...